Idem nefastos dies fastosque fecit, quia aliquando nihil cum populo agi utile futurum esset. He also established days on which no public business could be transacted and days on which it could, because sometimes nothing profitable ought to be done with people.

I'm having trouble with futurm esset. What construction is it? The closest thing I can find is the future active infinitive, but with that I've only ever seen the future active participle followed by esse.

It's a future participle of esse (futurum) followed by an imperfect subjunctive (esset), which amounts roughly to "it would be". Often in authentic Latin it's shortened to foret.

Since Latin doesn't technically have a future subjunctive, when the sense requires one it's generally made up of that future participle plus either the present or imperfect subjunctive of sum; the present subjunctive (futurus sim etc.) when the sequence of tenses requires it in an indirect question or some such, and the imperfect (futurus essem) when the sense of "would" is present - there is probably a more accurate technical description of that, but I can't think of one for the moment!

The subject of utile futurum esset is the infinitive phrase nihil cum populo agi; hence futurum and utile are neuter.

futurum esset is subjunctive because this is the reason that Numa gave or that was in his mind--not an explanation that the narrator (Livy) is giving--for Numa's action in setting fasti and nefasti days. It is "indirect speech" (oratio obliqua).